The Large-Number Limit for Reaction Networks (Part 2)

I’ve been talking a lot about ‘stochastic mechanics’, which is like quantum mechanics but with probabilities replacing amplitudes. In Part 1 of this mini-series I started telling you about the ‘large-number limit’ in stochastic mechanics. It turns out this is mathematically analogous to the ‘classical limit’ of quantum mechanics, where Planck’s constant goes to zero.

There’s a lot more I need to say about this, and lots more I need to figure out. But here’s one rather easy thing.

In quantum mechanics, ‘coherent states’ are a special class of quantum states that are very easy to calculate with. In a certain precise sense they are the best quantum approximations to classical states. This makes them good tools for studying the classical limit of quantum mechanics. As they reduce to classical states where, for example, a particle has a definite position and momentum.

We can borrow this strategy to study the large-number limit of stochastic mechanics. We’ve run into coherent states before in our discussions here. Now let’s see how they work in the large-number limit!

Coherent states

For starters, let’s recall what coherent states are. We’ve got different kinds of particles, and we call each kind a species. We describe the probability that we have some number of particles of each kind using a ‘stochastic state’. For starters, this is a formal power series in variables We write it as

where is an abbreviation for

But for to be a stochastic state the numbers need to be probabilities, so we require that

and

Sums of coefficients like this show up so often that it’s good to have an abbreviation for them:

Now, a coherent state is a stochastic state where the numbers of particles of each species are independent random variables, and the number of the th species is distributed according to a Poisson distribution.

Since we can pick ithe means of these Poisson distributions to be whatever we want, we get a coherent state for each list of numbers

What are the main properties of coherent states? For starters, they are indeed states:

More interestingly, they are eigenvectors of the annihilation operators

since when you differentiate an exponential you get back an exponential:

We can use this fact to check that in this coherent state, the mean number of particles of the th species really is For this, we introduce the number operator

where is the creation operator:

The number operator has the property that

is the mean number of particles of the th species. If we calculate this for our coherent state we get

Here in the second step we used the general rule

which is easy to check.

Rescaling

Now let’s see how coherent states work in the large-numbers limit. For this, let’s use the rescaled annihilation, creation and number operators from Part 1. They look like this:

Since

the point is that the rescaled number operator counts particles not one at a time, but in bunches of size For example, if is the reciprocal of Avogadro’s number, we are counting particles in ‘moles’. So, corresponds to a large-number limit.

To flesh out this idea some more, let’s define rescaled coherent states:

These are eigenvectors of the rescaled annihilation operators:

This in turn means that

Here we used the general rule

which holds because the ‘rescaled’ creation operator is really just the usual creation operator, which obeys this rule.

What’s the point of all this fiddling around? Simply this. The equation

says the expected number of particles of the th species in the state is if we count these particles not one at a time, but in bunches of size

A simple test

As a simple test of this idea, let’s check that as the standard deviation of the number of particles in the state goes to zero… where we count particle using the rescaled number operator.

A puzzle

I feel sure that in any coherent state, not only the variance but also all the higher moments of the rescaled number operators go to zero as Can you prove this?

Here I mean the moments after the mean has been subtracted. The th moment is then

I want this to go to zero as

Here’s a clue that should help. First, there’s a textbook formula for the higher moments of Poisson distributions without the mean subtracted. If I understand it correctly, it gives this:

Here

is the number of ways to partition an -element set into nonempty subsets. This is called Stirling’s number of the second kind. This suggests that there’s some fascinating combinatorics involving coherent states. That’s exactly the kind of thing I enjoy, so I would like to understand this formula someday… but not today! I just want something to go to zero!

If I rescale the above formula, I seem to get

We could plug this formula into

and then try to show the result goes to zero as But I don’t have the energy to do that… not right now, anyway!

Maybe you do. Or maybe you can think of a better approach to solving this problem. The answer must be well-known, since the large-number limit of a Poisson distribution is a very important thing.

I just feel it’s be good to know that all these higher moments go to zero at It’s obviously something we should expect! And it sounds like it could be a useful lemma in some calculations. So I was frustrated at first that it was hard to show. That indicated a weakness in my understanding. When it’s hard to show something obvious, it means you need to think more. So I thought more, and now I have found a much easier proof.

Should’t we first show that in the large number limit, goes to 0, and then think about the moments?

It appears that all cumulants of the poisson distribution are equal to the mean. That immediately gives you that the mean, variance, and third central moment are all and hence go to zero as does. In fact, all of the cumulants go to zero, which should imply that all of the central moments do since it should be possible to express central moments as polynomials in cumulants.

Okay, now I’m thinking that with your definitions, the order r cumulant will be , so that the mean is c, the variance is , and the third central moment is . So, all cumulants past the first go to zero. The translation to central moments past the first is not as quick as I first thought.

Thanks for telling me about cumulants. It would take me a while to get enough intuition for them to use them for something… though they vaguely remind me of ideas from statistical mechanics.

There are some funny formulas involved in translating between cumulants and moments… and I bet there’s some nice combinatorics lurking behind these. As I mentioned in my article, the moments of a Poisson distribution are somewhat complicated, involving Stirling numbers of the second kind. Are the cumulants simpler? If so, we might try to argue that the complexity of the moments of a Poisson distribution arises from translating cumulants into moments.

Yeah, there’s a close analogy. The cumulant generating function (CGF) is the log of the moment generating function (MGF). So, in the sense that the MGF is analogous to the partition function, the CGF is like the free energy. There’s a short discussion of this in the Wikipedia article on cumulants:

off the top of my head, can’t you just invoke the central limit theorem and state that for a large number of particles the Poisson distribution looks like a Gaussian with the correct mean and variance up to and then all moments become “classical” (whatever that means here)?

If I were smart enough maybe I could do what you’re saying. Can I use some version of the central limit theorem to prove that all the higher centered moments of a Poisson distribution with mean approach certain functions of as ?

Thanks, Dan—those remarks were more useful to me than the article on cumulants you pointed me to! I’m very used to the idea of differentiating a partition function to get useful information in physics: n-point functions, which are essentially moments. And I’m very used to the idea of taking the log of the partition function to get the free energy. But I’ve never thought much about differentiating the free energy! It’s obvious that these derivatives repackage the information in the derivatives of the partition function, but…

… oh, wait a minute. In quantum field theory, the log of the partition function is called the effective action, and its derivatives can be written as sums of connected Feynman diagrams just as the derivatives of the partition function can be expressed as sums of Feynman diagrams.

Actually this should explain the appearance of those Stirling numbers of the second kind. Just as any finite set can be written as the disjoint union of sets in a partition, any connected Feynman diagram can be written as a disjoint union of connected diagrams. The generating function for structures of any sort is the exponential of the generating function for ‘connected’ structures of that sort. I think I’m starting to get something here…

Glad to have helped, if only a little. Taking derivatives of the free energy is pretty common in statistical mechanics as well (or maybe I should say statistical thermodynamics). As I’m sure you know, the free energy is a thermodynamic potential that allows you to get pretty much all of the interesting thermodynamic quantities by taking appropriate derivatives. But, yeah, it is also used in quantum field theory by analogy. And there is certainly a lot of combinatorics hidden in there (which I don’t understand). The Wikipedia page mentions some of that as well:

Oh, and thanks for fixing my failed attempt at box quotes. How do I do those here?

Finally, for the sake of completeness and for anyone who might care, let me try to show the details of my claim that the cumulants of your rescaled number operator are . My thinking was that your rescaled coherent state is a Poisson distribution with mean . So, the ordinary number operator is a Poisson random variable in that distribution with cumulants . The rescaled number operator is just a scalar multiple of the ordinary number operator . So, the moment generating function of the rescaled number operator is (with a Poisson PMF with mean )

Excuse me, I retrying.
I write an idea that don’t work (too complex), but there are some strange intermediate results.
I am thinking that all is possible in Dirac notation (and in the second quantization for boson field):
and

but the commutator relations permit to exchange time the operator, and .

and the B bunch number operator can be:

so that:

some solutions are simple, for example (the second is obtainable with Mathematica):

I tried to fix your comment, but I don’t understand it so it’s hard to fix. Your equations contained strange expressions like ===, etcetera. In general it’s better to explain things in words and avoid unusual and unexplained mathematical symbols.

Excuse me, I did not want to persevere in the error, try a different source.
I re-re-write an idea that don’t work (too complex), but there are some strange intermediate results.
I am thinking that all is possible in Dirac notation (and in the second quantization for boson field):
and
in Dirac notation the renormalization is:
it is possible to obtain the number of particle for the distribution function:
using the commutation relations:
it is possible to obtain the derivative of the distribution operator:
the commutatior relation permit to exchange time the operator, and , so that:
I try a B bunch number operator (here I am not sure of the definition, Q can be a normalization):
and the square of the number operator:
so that:
or:
some solutions are simple, for example (the second is obtainable with Mathematica):

I am thinking a simple idea, and monstrous calculations.
If a system have a operator distribution that depend on the temperature, then it is possible to apply the statistical mechanic over the second quantization.
If this happen there is a connection between Feynmann diagram and chemical reaction (from elementary particles interaction to molecule chemical reaction potential).
Can the Feynmann diagram be applied to the molecule reaction using simply an approximation of the reaction potential instead of the potential between elementary particles?

Okay, John, you have a perfectly good answer to your puzzle, but here’s an attempt at an abstract nonsense proof based on cumulants (for my own personal edification). I’ll start with some overly complicated notation (a prerequisite for any abstract nonsense proof). All of this is in the context of the PMF

where the mean is . We have several random variables to worry about. The ordinary number operator has MGF:

The rescaled number operator has MGF:

Finally, we have the -centered, rescaled number operator which has MGF:

Now, the CGF of the -centered, rescaled number operator is

So, the first cumulant of is

where we recall that . For higher order cumulants, the extra term gets killed and we get the same answer as for the rescaled number operator, i.e., for we have that

Thus, all cumulants for the centered, rescaled number operator of order greater than are proportional to and the first cumulant is zero. Furthermore, the moments of the centered, rescaled number operator can be written as polynomials of degree 1 or greater in its cumulants. Therefore, all of the -centered moments of the rescaled number operator are either zero or proportional to and hence go to zero in the limit.

I’ve always been partial to direct demonstration proofs, so I like your proof better. But I’ve already spent so much time on this that I felt I should finish it…. Now I’d better get back to the work I’m being paid for. :)

If I ever meet you I will gladly buy you a beer or two—or any beverage of your choice. That may recompense you for your unpaid work (though I know you did it just for fun).

I like your approach, because it illustrates a little bit of the power and beauty of cumulants. First, the cumulant generating function of the Poisson distribution is so much simpler than the moment generating function. Second, it’s nice how all the higher cumulants don’t care when you translate the Poisson distribution to make its mean zero. (I bet that’s a general property of cumulants but I’m too lazy to think about it now.)

Your approach is also a bit like my final approach, in the following sense. We avoid working directly with moments and work with other quantities, of which the moments are certain polynomial functions. For me, I happened to notice that when

is a Poisson distribution, these quantities

are simpler than the moments:

They obey

Someone who knew more about Poisson distributions would presumably know all sorts of tricks like this, including cumulants, but I’ve never really studied them before.

So, your comment got me to thinking about the relationship between falling powers moments (what seem to be referred to as “factorial moments” elsewhere) and cumulants. And it sent me down a rabbit hole called the Umbral Calculus, which I had never heard of before. If you haven’t either, then the introduction of this survey is nice:

Using the umbral calculus, it details the relationships between moments, factorial moments, central moments, and cumulants of random variables. The Poisson distribution seems to come up a lot. There’s a lot of notation and (unfortunately) I don’t really have the time to understand it fully. But here’s a few tidbits I pulled out that might be relevant to the present discussion:

1. Proposition 7.1 shows that the factorial moment generating function is where is the ordinary moment generating function. So, for the Poisson with mean , we’d get demonstrating the property you used in your proof above, i.e., the the r-th factorial moment of the Poisson is just .

2. Footnote 3 is a quote giving a short history of cumulants, which I found interesting.

3. The first sentence of section 4 says:

The family of Poisson r.v.’s plays a crucial role in the theory of r.v.’s, especially because the most general infinitely [sic] distribution may be represented as a limit of an appropriate sequence of compound Poisson processes (cf. [4]).

Now, [4] is the second volume of Feller’s classic treatise on probability theory. I’m almost ashamed to admit that I don’t have access to a copy of that text, so I’m not really sure what that statement means, but it sounds like it might be relevant to your classical limit, no? After all, you understand the limit for Poisson distributions….

it seems that the theorem in Feller says something about all infinitely divisible distributions on the nonnegative integers being compound Poisson, meaning they can be represented as

where is Poisson and are iid and independent of . And infinitely divisible seems to be a condition on the characteristic function of the distribution. It looks like it means that all roots of the characteristic function are themselves characteristic functions.

How To Write Math Here:

You need the word 'latex' right after the first dollar sign, and it needs a space after it. Double dollar signs don't work, and other limitations apply, some described here. You can't preview comments here, but I'm happy to fix errors.